channel_metrics.channel_fidelity

Computes the channel fidelity between two quantum channels.

Functions

channel_fidelity(choi_1, choi_2)

Compute the channel fidelity between two quantum channels [1].

Module Contents

channel_metrics.channel_fidelity.channel_fidelity(choi_1, choi_2)

Compute the channel fidelity between two quantum channels [1].

Let \(\Phi : \text{L}(\mathcal{Y}) \rightarrow \text{L}(\mathcal{X})\) and \(\Psi: \text{L}(\mathcal{Y}) \rightarrow \text{L}(\mathcal{X})\) be quantum channels. Then the root channel fidelity defined as

\[\sqrt{F}(\Phi, \Psi) := \text{inf}_{\rho} \sqrt{F}(\Phi(\rho), \Psi(\rho))\]

where \(\rho \in \text{D}(\mathcal{Z} \otimes \mathcal{X})\) can be calculated by means of the following semidefinite program (Proposition 50) in [1],

\[\begin{split}\begin{align*} \text{maximize:} \quad & \lambda \\ \text{subject to:} \quad & \lambda \mathbb{I}_{\mathcal{Z}} \leq \text{Re}\left( \text{Tr}_{\mathcal{Y}} \left( Q \right) \right),\\ & \begin{pmatrix} J(\Phi) & Q^* \\ Q & J(\Psi) \end{pmatrix} \geq 0 \end{align*}\end{split}\]

where \(Q \in \text{L}(\mathcal{Z} \otimes \mathcal{X})\).

Examples

For two identical channels, we should expect that the channel fidelity should yield a value of \(1\).

>>> import numpy as np
>>> from toqito.channels import dephasing
>>> from toqito.channel_metrics import channel_fidelity
>>>
>>> # The Choi matrices of dimension-4 for the dephasing channel
>>> choi_1 = dephasing(4)
>>> choi_2 = dephasing(4)
>>> np.around(channel_fidelity(choi_1, choi_2), decimals=2)
np.float64(1.0)

We can also compute the channel fidelity between two different channels. For example, we can compute the channel fidelity between the dephasing and depolarizing channels.

>>> import numpy as np
>>> from toqito.channels import dephasing, depolarizing
>>> from toqito.channel_metrics import channel_fidelity
>>>
>>> # The Choi matrices of dimension-4 for the dephasing and depolarizing channels
>>> choi_1 = dephasing(4)
>>> choi_2 = depolarizing(4)
>>> np.around(channel_fidelity(choi_1, choi_2), decimals=2)
np.float64(0.5)

References

[1] (1,2,3)

Vishal Katariya and Mark M. Wilde. Geometric distinguishability measures limit quantum channel estimation and discrimination. Quantum Information Processing, Feb 2021. URL: http://dx.doi.org/10.1007/s11128-021-02992-7, doi:10.1007/s11128-021-02992-7.

Raises:

  • ValueError – If matrices are not of equal dimension.

  • ValueError – If matrices are not square.

Parameters:

  • choi_1 (numpy.ndarray) – The Choi matrix of the first quantum channel.

  • choi_2 (numpy.ndarray) – The Choi matrix of the second quantum channel.

Returns:

The channel fidelity between the channels specified by the quantum channels corresponding to the Choi matrices choi_1 and choi_2.

Return type:

float